Hydrodynamics of bacteriophage migration along bacterial flagella

Panayiota Katsambaa and Eric Lauga†

Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge CB3 0WA, United Kingdom

(Dated: November 2, 2018)

Bacteriophage viruses, one of the most abundant entities in our planet, lack the ability to moveindependently. Instead, they crowd fluid environments in anticipation of a random encounter witha bacterium. Once they ‘land’ on the cell body of their victim, they are able to eject their geneticmaterial inside the host cell. Many phage species, however, first attach to the flagellar filamentsof bacteria. Being immotile, these so-called flagellotropic phages still manage to reach the cellbody for infection, and the process by which they move up the flagellar filament has intrigued thescientific community for decades. In 1973, Berg and Anderson (Nature, 245, 380-382) proposedthe nut-and-bolt mechanism in which, similarly to a rotated nut that is able to move along a bolt,the phage wraps itself around a flagellar filament possessing helical grooves (due to the helicalrows of flagellin molecules) and exploits the rotation of the flagellar filament in order to passivelytravel along it. One of the main evidence for this mechanism is the fact that mutants of bacterialspecies such as Escherichia coli and Salmonella typhimurium that possess straight flagellar filamentswith a preserved helical groove structure can still be infected by their relative phages. Using twodistinct approaches to address the short-range interactions between phages and flagellar filaments,we provide here a first-principle theoretical model for the nut-and-bolt mechanism applicable tomutants possessing straight flagellar filaments. Our model is fully analytical, is able to predict thespeed of translocation of a bacteriophage along a flagellar filament as a function of the geometryof both phage and bacterium, the rotation rate of the flagellar filament, and the handedness of thehelical grooves, and is consistent with past experimental observations.

aCurrent address: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK† e.lauga@damtp.cam.ac.uk

2

Head

Collar

Tail

Long Tail Fibres

Base Plate

DNA

Protein

2D 3D

FIG. 1. Bacteriophages: (A) A typical morphology of a bacteriophage, such as the Enterobacteria T4 phage (Adenosine,Wikimedia Commons); (B) Electron micrograph of bacteriophages attached to a bacterial cell, (Dr. G. Beards, WikimediaCommons); (C-E) Phages can come in various shapes: (C) Myoviridae; (D) Podoviridae; (E) Siphoviridae [8]. Panels C-E:Reprinted by permission from Suttle CA, “Viruses in the sea”, Nature, 437 (356), 356-361, Copyright 2005 Springer Nature.

I. INTRODUCTION

As big as a fraction of a micrometre, bacteriophages (in short phages), are ‘bacteria-eating’ viruses (illustrated inFig. 1) that infect bacteria and replicate within them [1]. With their number estimated to be of over 1031 on theplanet, phages are more abundant than every other organism on Earth combined [2–8].

Phages have been used extensively in genetic studies [1, 4, 9], and their future use in medicine is potentially of evengreater impact. The global rise in antibiotic resistance, as reported by the increasing number of multidrug-resistantbacterial infections [10], poses one of the greatest threats to human health of our times, and phages could offer thekey to resolution. Indeed, phages have been killing bacteria for way longer than humanity has been fighting againstbacterial infections, with as many as 1029 infections of bacterial cells by oceanic phages taking place every day [11, 12].Phage therapy is an alternative to antibiotics that has been used for almost a century and offers promising solutionsto tackle antibiotic-resistant bacterial infections [13]. Furthermore, the unceasing phage-bacteria war taking place inenormous numbers offers the scientific community great opportunities to learn. For example, the ability of phagesto update their infection mechanisms in response to bacterial resistance could offer us valuable insight into updatingantibiotics treatment against multi drug-resistant pathogenic bacteria [14]. In addition, the high selectivity of theattachment of a phage to the receptors on the bacterial cell surface and the species it infects could help identify possibletarget points of particular pathogenic bacteria for drugs to attack [15]. In general, extensive studies of bacteriophageinfection strategies could not only reveal vulnerable points of bacteria, but may help uncover remarkable biophysicalphenomena taking place at these small scales.

Infection mechanisms can vary across the spectrum of phage species [15, 16]. Lacking the ability to move indepen-dently, phages simply crowd fluid environments and rely on a random encounter with a bacterium in order to landon its surface and accomplish infection using remarkable nanometre size machinery. Typically, the receptor-bindingproteins located on the long tail fibres recognise and bind to the receptors of the host cell via a two-stage processcalled phage adsorption [15]. The first stage is reversible, and is followed by irreversible attachment onto the cellsurface. Subsequently, the genetic material is ejected from their capsid-shaped head, through their tail, which is ahollow tube, into the bacterium [17, 18].

While all phages need to find themselves on the surface of the cell body for infection to take place, there is aclass of phages, called flagellotropic phages, that first attach to the flagellar filaments of bacteria. Examples includethe χ-phage infecting Escherichia coli (E. coli) and Salmonella typhimurium (Salmonella), the phage PBS1 infectingBacillus subtilis (B. subtilis) and the recently discovered phage vB VpaS OWB (for short OWB) infecting Vibrioparahaemolyticus (V. parahaemolyticus) [19], illustrated in Fig. 2.

Given the fact that phages are themselves incapable of moving independently and that the distance they would haveto traverse along the flagellar filament is large compared to their size, they must find an active means of progressingalong the flagellar filament. In Ref. [22], electron microscopy images of the flagellotropic χ-phage, shown in Figs. 2Cand D, were provided to show that the mechanism by which χ-phage infects E. coli consists of travelling along theoutside of the flagellar filament until it reaches the base of the flagellar filament where it ejects its DNA.

A possible mechanism driving the translocation of χ-phage along the flagellar filament was first proposed in Bergand Anderson’s seminal paper as the ‘nut-and-bolt’ mechanism [23]. Their paper is best known for establishing thatbacteria swim by rotating their flagellar filaments. One of the supporting arguments was the proposed mechanism

3

FIG. 2. Flagellotropic phages: (A) Attachment of phage OWB to V. parahaemolyticus [19]. Red arrows indicate phageparticles. (B) Phage PBS1 adsorbed to the flagellar filament of a B. Subtilis bacterium with its tail fibres wrapped around theflagellar filament in a helical shape with a pitch of 35 nm [20]. The phage hexagonal head capsid measures 120 nm from edgeto edge [21]. Reprinted (amended) by permission from American Society for Microbiology from Raimondo LM, Lundh NP,Martinez RJ, “Primary Adsorption Site of Phage PBS1: the Flagellum of Bacillus”, J. Virol., 1968, 2 (3), 256-264, Copyright1968, American Society for Microbiology. (C) χ-phage of E. coli [22]. The head measures 65 to 67.5 nm between the parallelsides of the hexagon [22]; (D) χ-phage at different times between attachment on the flagellar filament of E. coli and reachingthe base of the filament [22]. Arrows point to the bases of the flagella. Panels C-D: Reprinted (amended) by permission fromAmerican Society for Microbiology from Schade SZ, Adler J, Ris H, “How Bacteriophage χ Attacks Motile Bacteria”, J. Virol.,1967, 1 (3), 599-609, Copyright 1967, American Society for Microbiology.

where the phage plays the role of the nut and the bolt is the flagellar filament, with the grooves between the helicalrows of flagellin molecules making up the flagellar filament serving as the threads [23] (Figs. 3A and B). A phagewould then wrap around the flagellar filament and the rotation of the latter would result in the translocation of thephage along it.

A mutant of Salmonella that has straight flagellar filaments, but possesses the same helical screw-like surface dueto the arrangement of the flagellin molecules [24] is non-motile due to the lack of chiral shape yet fully sensitive toχ-phage [26], i.e. the phages manage to get transported to the base of the flagellar filament. This is consistent withthe nut-and-bolt mechanism and was used as evidence that the flagellar filament is rotating [23].

More evidence in support of the nut-and-bolt mechanism were provided 26 years after its inception in a work studyingstrains of Salmonella mutants with straight flagellar filaments whose motors alternate from rotating clockwise (CW)and counter-clockwise (CCW) [27]. The directionality of rotation is crucial to the mechanism as CCW rotationwill only pull the phage toward the cell body if the phage slides along a right-handed groove. In order to test thedirectionality, the authors used a chemotaxis signalling protein that interacts with the flagellar motor, decreasing theCCW bias. They found that strains with a large CCW bias are sensitive to χ-phage infection, whereas those withsmall CCW bias are resistant, in agreement with the proposed nut-and-bolt mechanism.

Details of the packing of the flagellin molecules that give rise to the grooves can be found in Ref. [25] and examplesare shown in Figs. 3A and B. It is important to note that the packing of flagellin molecules produces two overlappingsets of helical grooves, a long-pitch and a short-pitch set of grooves which are of opposite chirality [24]. Once incontact with a rotating flagellar filament, it is anticipated that the phage fibres will wrap along the short-pitchgrooves. Indeed, the findings of Ref. [27] show that the directionality of phage translocation correlates with thechirality of the short-pitch grooves.

The flagellar filaments of bacteria can take one of the twelve distinct polymorphic shapes as illustrated in Figs. 3Band C. The authors in Ref. [27] examined flagellar filaments with different polymorphic forms, since the differentarrangements of the flagellin subunits give rise to grooves with different pitch and chirality [25], as shown in Fig. 3B.The L-type straight flagellar filament (f0) that was used by Ref. [27] has both left-handed long-pitch grooves and right-handed short-pitch grooves [24]. Given that the short-pitch grooves are relevant to the wrapping of the fibres, thefindings of Ref. [27] that bacteria with their flagellar filament in the f0 polymorphic state (i.e. with right-handed short-pitch grooves) and with a large CCW bias are sensitive to χ-phage infection, are in agreement with the nut-and-bolt

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FIG. 3. Bacterial flagellar filaments and polymorphism: (A) Structure of straight flagellar filament from a mutant ofSalmonella typhimurium [24]. Reprinted with permission from O’Brien EJ, Bennett PM, “Structure of straight flagella from amutant Salmonella”, J. Mol. Biol., 70 (1), 145-152, Copyright 1972 Elsevier. The L-type straight flagellar filament (left) hastwo types of helical grooves, left-handed long-pitch and right-handed short-pitch grooves, whereas the R-type straight filament(right) has right-handed long-pitch and left-handed short-pitch grooves. Examples of short-pitch grooves are marked by thinblue lines and indicated by arrows. (B) Schematic of some polymorphic states of the flagellar filament, from left to right: L-typestraight, normal (left-handed shape), curly (right-handed) and R-type straight. The top panel shows the shape of the filamentswhile the bottom panel displays the arrangements of flagellin subunits. Examples of short-pitch grooves are marked by thinblue lines and indicated by arrows [25]. Reprinted with permission from Namba K, Vonderviszt F, “Molecular architecture ofbacterial flagellum”, Q. Rev. Biophys., 1997, 30 (1), 1-65, Copyright 1997 Cambridge University Press.

mechanism.The same study also argued that the translocation time of the phage to the cell body is less than the flagellar

filament reversal interval, a necessary condition for successful infection by the virus of wild-type bacteria whosemotors alternate between CCW and CW rotation. Their estimated translocation speeds on the order of microns perseconds give a translocation time which is less than the CCW time interval of about a 1 s [27].

Relevant to the nut-and-bolt mechanism are also the findings of an alternative mechanism for adsorption of theflagellotropic phages φCbK and φCb13 that interact with the flagellar filament of Caulobacter crescentus using afilament located on the head of the phage [28], instead of the tail or tail fibres that other flagellotropic phages use,such as χ and PBS1. This study also reports on a higher likelihood of infection with a CCW rotational bias thatis consistent with the nut-and-bolt mechanism. Notably, phages can also attach to curli fibres, which are bacterialfilaments employed in biofilms; however due to the lack of helical grooves and rotational motion of these filaments,phages are unable to move along them [29].

In this paper, we theoretically examine the nut-and-bolt mechanism from a quantitative point of view and performa detailed mathematical analysis of the physical mechanics at play. We focus on the virus translocation along straightflagellar filaments in mutants such as the mutant of Salmonella used in Ref. [23]. A flagellotropic phage can wraparound a given flagellar filament using its tail fibres (fibres for short), its tail, or in some cases a filament emanatingfrom the top of its head [28] and the models we develop can address all these relevant morphologies. A schematicdiagram of the typical geometry we consider is shown in Fig. 4.

A phage floating in a fluid whose fibres suddenly collide with a flagellar filament rotating at high frequency willundergo a short, transient period of wrapping, during which the length of the fibres that are wrapped around thefilament is increasing. In this paper we study the translocation of the phage once it has reached a steady, post-wrappingstate, and assume that it is moving rigidly with no longer any change in the relative virus-filament configuration.

In order to provide first-principle theoretical modelling of the nut-and-bolt mechanism, we build in the paper ahierarchy of models. In §II, we start with a model of drag-induced translocation along smooth flagellar filaments thatignores the microscopic mechanics of the grooves yet implicitly captures their effect by coupling the helical shapeof the fibres with anisotropy in motion in the local tangent plane of the flagellar filament. Having acquired insightinto the key characteristics of the mechanism, we proceed by building a refined, more detailed model of the guided

5

FIG. 4. Schematic model of the translocation of a flagellotropic phage along the straight flagellar filament ofa mutant bacterium. The phage, illustrated in dark green, is wrapped around the straight flagellar filament (light bluecylinder) using its fibres, with its tail and head protruding in the bulk fluid.

translocation of phages along grooved flagellar filaments by incorporating the microscopic mechanics of the grooves in§III. This is done by including a restoring force that acts to keep the fibres in the centre of the grooves, thereby guidingtheir motion, as well as a resistive force acting against the sliding motion. In both models, we proceed by consideringthe geometry, and the forces and torques acting on the different parts of the phage. We use the resistive-force theoryof viscous hydrodynamics in order to model the tail and tail fibres which are both slender [30, 31]. The portion ofthe phage wrapping around the flagellar filament is typically the fibres. They experience a hydrodynamic drag fromthe motion in the proximity of the rotating flagellar filament along which they slide in the smooth flagellum model,or a combination of a guiding and resistive forces in the grooved flagellum model. Parts sticking out in the bulk awayfrom the flagellar filament experience a hydrodynamic drag due to their motion in an otherwise stagnant fluid.

We build in our paper a general mathematical formulation relevant to a broad phage morphology. In our typicalgeometry of phages wrapping around flagellar filaments using their fibres, two limits arise for long-tailed and short-tailed phages. Long-tailed phages have their tail and head sticking out in the bulk, away from the flagellar filament,whereas for short-tailed phages only the head is exposed to the bulk fluid. The hydrodynamic torque actuating thetranslocation is provided by the parts sticking out in the bulk.

We compare the results from the two models addressing the two geometrical limits and find these to be consistentwith each other and with the predictions and experimental observations of Refs. [23, 27]. In particular, we predictquantitatively the speed of phage translocation along the flagellar filament they are attached to, and its criticaldependence on the interplay between the chirality of the wrapping and the direction of rotation of the filament, aswell as the geometrical parameters. Most importantly we show that our models capture the correct directionality oftranslocation, i.e. that CCW rotation will only pull the phage toward the cell body if the phage slides along a right-handed groove, and predict speeds of translocation on the order of µms−1, which are crucial for successful infectionin the case of bacteria with alternating CCW and CW rotations.

II. DRAG-INDUCED TRANSLOCATION ALONG SMOOTH FLAGELLAR FILAMENTS

II.1. Geometry

As our first model, we consider the flagellar filament as a straight, smooth rod aligned with the z-axis and of radiusRfl. The phage has a capsid head of size 2ah, a tail of length Lt and fibres that wrap around the flagellar filament.We implicitly capture the effect of the grooves (i) by imposing that the fibres that emanate from the bottom of thetail of the phage are wrapped around the flagellar filament in a helical shape and (ii) via the anisotropy in the dragarising from the relative motion between the fibres and the rotating flagellar filament. The helical shape of the fibreshas helix angle α, as shown in Fig. 5. With the assumption that the gap between the fibres and the flagellar filamentis negligible compared to the radius Rfl of the flagellar filament, the centreline rfib(s) of the fibres, parametrised by

6

FIG. 5. Mathematical model of drag-induced translocation along a smooth flagellar filament. The fibres experiencean anisotropic drag due to their motion in the proximity of the rotating flagellar filament. The phage, shown in green, hasa capsid head of size 2ah, a tail of length Lt, fibres of helical shape with helix angle α and cross-sectional radius rfib, and istranslocating along a straight flagellar filament, shown in light blue, of radius Rfl. The flagellar filament is rotating at a rateωfl. The phage is translocating at speed U and rotating with rate ωp about the flagellar filament. The inset shows a shortsegment of the phage fibre at a distance d from the local tangent plane of the flagellar filament.

the contour length position s, is described mathematically as

rfib(s) =

(Rfl cos

(s

Rfl/ sinα

), hRfl sin

(s

Rfl/ sinα

), s cosα

), −LL

fib < s < LRfib, (1)

with total contour length Lfib = LLfib + LR

fib, where we allow for fibres extending to both sides of the base of the tail

to have lengths LLfib (left side) and LR

fib (right side). The helix wrapping is right-handed or left-handed according towhether the chirality index h takes the value +1 or −1 respectively.

Assuming the phage to move rigidly and working in the laboratory frame, every point r on the phage moves withvelocity Uez + ωpez ∧ r. The flagellar filament is assumed to rotate at rate ωfl along its axis, and thus its velocityis given by ωflez ∧ r in a fluid that is otherwise stationary, where the value of ωfl is known. The purpose of ourcalculation is to compute the two unknown quantities, U and ωp, in terms of ωfl by enforcing the overall force andtorque balance on the phage along the z-axis.

II.2. Forces and moments

In order to calculate the forces and torques acting on the tail and fibres we use the resistive-force theory of viscoushydrodynamics (RFT in short) [30, 31]. This theoretical framework predicts the viscous tractions due to the motion ofa slender filament in a viscous fluid by integrating fundamental solutions of the Stokes equations of hydrodynamics [32]along the centreline of the filament.

In an infinite fluid, the instantaneous hydrodynamic force per unit length exerted on a filament due to its motionin an otherwise stagnant viscous fluid is given by

f(s) = −ζ⊥[v(s)−

(∂r

∂s

∣∣∣∣s

.v(s)

)∂r

∂s

∣∣∣∣s

]− ζ‖

(∂r

∂s

∣∣∣∣s

.v(s)

)∂r

∂s

∣∣∣∣s

, (2)

where ∂r∂s

∣∣s

and v(s) are the local unit tangent and velocity of the filament relative to the fluid at contour-lengthposition s respectively, and ζ‖, ζ⊥ are the drag coefficients for motion parallel and perpendicular to the local tan-gent [30, 33]. For a slender rod of length L and radius r in an infinite fluid, we have

ζ⊥ ≈4πµ

ln (L/r), ρ ≡ ζ‖/ζ⊥ ≈ 1/2, (3)

where µ is the dynamic viscosity of the fluid.

7

The fact that the perpendicular drag coefficient is twice the parallel one captures the fact that it is twice as hardto pull a rod through a viscous fluid in a direction perpendicular to its length than lengthwise. This drag anisotropyis at the heart of the propulsion physics for microorganisms such as bacteria and spermatozoa [31].

As a result, the total hydrodynamic force and the z-component of the torque on the phage tail due to its motionin the fluid are given by

Ftail = −Lt∫0

[ζ‖,tttailttail + ζ⊥,t(1− ttailttail)

]· urel

tail(s) ds, (4)

ez ·Mtail = −ez ·Lt∫0

rt(s) ∧[ζ‖,tttailttail + ζ⊥,t(1− ttailttail)

]· urel

tail(s)

ds, (5)

where rt(s) and ttail(s) are the position and tangent vectors of the fibre centreline at contour length position srespectively. The symbols ζ⊥,t, ζ‖,t are the drag coefficients for motion perpendicular and parallel to the local tangent,with ζ‖,t ≡ ρtζ⊥,t and the velocity of the tail relative to the fluid is

ureltail(s) = ωp (ez ∧ rt) + Uez. (6)

For the fibres, we use the version of RFT modified to capture the motion of slender rods near a surface. The flagellarfilament is rotating at rate ωfl, thus the velocity of the fibres relative to the flagellar filament is given by

urelfib(s) = Ωrel (ez ∧ rfib) + Uez, (7)

where the relative angular velocity is given by

Ωrel = ωp − ωfl. (8)

The expressions for the fibres are similar, and we have

Ffib = −

L(R)fib∫

−L(Lfib)

[ζ‖,fibtfibtfib + ζ⊥,fib(1− tfibtfib)

]· urel

fib(s) ds, (9)

ez ·Mfib = −ez ·

L(R)fib∫

−L(L)fib

rfib(s) ∧[ζ‖,fibtfibtfib + ζ⊥,fib(1− tfibtfib)

]· urel

fib(s)

ds. (10)

The difference between the expressions in Eqs. 4,5 and Eqs. 9,10 is that in the latter we use the appropriateresistance coefficients, ζ⊥,fib and ζ‖,fib, for motion at a small, constant distance d from a nearby surface, in directionsperpendicular and parallel to the local tangent of the fibre respectively, given by

ζ⊥,fib ≈4πµ

ln (2d/rfib), (11)

with ζ‖,fib = ρfibζ⊥,fib and again ρfib ≈ 1/2 (see Ref. [34] and references therein). These results are valid in the limitin which the distance d between the fibre and the surface of the flagellar filaments is much smaller than the radius ofthe flagellar filament (d Rfl), such that the surface of the smooth flagellar filament is locally planar. Importantly,the very drag anisotropy that allows the rotation of helical flagellar filaments to propel bacteria in the bulk will alsoenable the rotation of helical fibres around a smooth filament to lead to translocation along the axis of the filament.

If the tail is straight with total length Lt and the head is spherical with radius ah, the position of the centre of thehead is given by

rh = rb + Ltttail + ahth, (12)

where rb = (Rfl, 0, 0) is the base of the tail from which the fibres emanate. The drag force and torque due to themotion of the head in the otherwise stagnant fluid are given by

Fhead = −6πµahurelhead, (13)

ez ·Mhead = ez ·[−6πµah

(rh ∧ urel

head

)]− 8πµa3hωp, (14)

8

with urelh given by

urelh = ωp (ez ∧ rh) + Uez. (15)

Taking th = ttail, the centre of the head will be located at position rh = rb + (Lt + ah) ttail. Evaluating theintegrals in Eqs. 4,5,9 and 10 and the expressions of Eqs. 13,14 with this geometry we obtain the forces and torquesexerted on the different parts of the phage (projected along the z-axis),

ez · Ffib = −ζ⊥,fibLfib

[(sin2 α+ ρfib cos2 α)U − (1− ρfib) sinα cosαhΩrelRfl

], (16)

ez ·Mfib = −hζ⊥,fibRflLfib

[hRflΩrel(cos2 α+ ρfib sin2 α)− (1− ρfib) sinα cosαU

], (17)

ez · Ftail = −ζ⊥,tLt

U[1− (1− ρt)t2z

]− ωpRfl(1− ρt)tytz

, (18)

ez ·Mtail = −ζ⊥,tωp

[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)

]− (1− ρt)ty(Utz + ωpRflty)RflLt

, (19)

ez · Fhead = −6πµahU, (20)

ez ·Mhead = −6πµahωp

[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]− 8πµa3hωp, (21)

where we use the notation ttail = (tx, ty, tz) for the components of the tangent of the tail.

II.3. Phage translocation: General formulation

The overall force and torque balance on the phage along the z-axis is written as

0 = ez · [Ffib + Ftail + Fhead], 0 = ez · [Mfib + Mtail + Mhead], (22)

which leads to

0 =− ζ⊥,fibLfib

[(sin2 α+ ρfib cos2 α)U − (1− ρfib) sinα cosαhΩrelRfl

]− ζ⊥,tLt

U[1− (1− ρt)t2z

]− ωpRfl(1− ρt)tytz

− 6πµahU, (23)

0 =− hζ⊥,fibRflLfib

[hRflΩrel(cos2 α+ ρfib sin2 α)− (1− ρfib) sinα cosαU

]− ζ⊥,t

ωp

[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)

]− (1− ρt)ty(Utz + ωpRflty)RflLt

− 6πµahωp

[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]− 8πµa3hωp. (24)

Writing this system in a formal matrix form for U and ωp in terms of ωfl gives(A BB D

)(Uωp

)=

(ZH

)ωfl, (25)

where we have

A = ζ⊥,tLt

[1− (1− ρt)t2z

]+ ζ⊥,fibLfib(sin

2 α+ ρfib cos2 α) + 6πµah, (26)

B = − [ζ⊥,tLt(1− ρt)tytz + hζ⊥,fibLfib(1− ρfib) sinα cosα]Rfl, (27)

D = ζ⊥,tLt

[R2

fl + LtRfltx +L2t

3(1− t2z)− (1− ρt)R2

flt2y

]+ ζ⊥,fibLfibR

2fl(cos2 α+ ρfib sin2 α)

+ 6πµah[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(1− t2z)]

+ 8πµa3h, (28)

Z = −hζ⊥,fibLfibRfl(1− ρfib) sinα cosα, (29)

H = ζ⊥,fibLfibR2fl(cos2 α+ ρfib sin2 α). (30)

9

Inverting Eq. 25 gives the translation linear and rotational speeds as(Uωp

)=

1

AD −B2

(D −B−B A

)(ZH

)ωfl. (31)

The full expressions for (DZ − BH),(AD − BC) and (−CZ + AH) for a general phage geometry are given in theSupplementary Material (see [35]).

II.4. Two limits: long vs short-tailed phages

From the experimental images in Fig. 2 we can distinguish two geometries of wrapping according to how far thetail and head are sticking out in the bulk fluid and away from the flagellar filament. We thus proceed by consideringthe two limiting geometries of long- and short-tailed phages.

II.4.1. Long-tailed phages

Examples of long-tailed morphology include the χ-phage of E. coli and the PBS1 phage of B. subtilis shown inFigs. 2B and D. We use below the χ-phage as a typical long-tailed phage, whose detailed dimensions are reported inRef. [22]. The hexagonal head measures 650− 675 A between parallel sides (that is 2ah ≈ 650− 675 A). The tail is aflexible rod that is 2, 200 A long and 140 A wide, and the tail fibres are 2, 000−2, 200 A long and 20−25 A wide. Theflagellar filaments are 5 − 10 µm long and have a diameter 20 nm, hence Rfl ≈ 100 A [36]. For this specific phage,we thus have ah ≈ 30 nm, Lt ≈ 220 nm, Lfib ≈ 200 nm, rfib ≈ 1 nm and Rfl ≈ 10 nm. From this we see that we cansafely assume that Rfl, ah Lt, Lfib. Notice however that Lfib ≈ Lt and that Rfl and ah are of the same order ofmagnitude. Our variables are thus divided into the short lengthscales of ah, Rfl and the long lengthscales of Lfib, Lt.With these approximations we obtain the translocation speed as

Ulong ≈− hωflRfl(1− ρfib) sinα cosα Glong, (32)

Glong =ζ⊥,fibLfib

[13ζ⊥,tLt(1− t2z) +

[ζ⊥,tRfltx + 6πµah(1− t2z)

]]13ζ⊥,tLt(1− t2z)

[ζ⊥,tLt [1− (1− ρt)t2z] + ζ⊥,fibLfib(sin

2 α+ ρfib cos2 α)

] (33)

≈ ζ⊥,fibLfib[ζ⊥,tLt [1− (1− ρt)t2z] + ζ⊥,fibLfib(sin

2 α+ ρfib cos2 α)

] , (34)

with a relative error of O (ah/Lt, ah/Lfib, Rfl/Lt, Rfl/Lfib). Details of the approximation are given in the Supple-mentary Material (see [35]).

II.4.2. Short-tailed phages

Phages with very short tails that use their fibres to wrap around flagellar filaments are equivalent geometrically tophages that use their entire tail for wrapping since in both cases there is a filamentous part of the phage wrappedaround the flagellar filament and the head is sticking out in the bulk close to the surface of the filament. For example,the phage OWB that infects V. parahaemolyticus studied in Ref. [19] and shown in Fig. 2A, uses its tail for wrapping.

In order to avoid any confusion, we will carry out the calculations of this section using the geometry of short-tailedphages, and assume that (i) the tail is negligible and (ii) the fibres are wrapping around the flagellar filament. In thiscase we obtain a translocation speed of

Ushort = −hRflωfl(1− ρfib) sinα cosα Gshort, (35)

Gshort =ζ⊥,fibLfib6πµah

[(Rfl + ahtx)2 + a2ht

2y + 4

3a2h

][ρfibζ

2⊥,fibL

2fibR

2fl + 6πµahζ⊥,fibLfibR

2fl(cos2 α+ ρfib sin2 α)

+6πµah[ζ⊥,fibLfib(sin

2 α+ ρfib cos2 α) + 6πµah] [

(Rfl + ahtx)2 + a2ht2y + 4

3a2h

]] , (36)

with details of the calculation given in the Supplementary Material (see [35]).Note that the results for phages which use their tail to wrap around the flagellar filament can be readily obtained

by replacing Lfib with Lt and all relevant quantities in the above result.

10

II.4.3. Interpretation and discussion of the asymptotic results

We now interpret and compare the results we obtained in Eqs. 32 and 35. As we now see, our formulae give the cor-rect directionality and speed of translocation in agreement with the qualitative predictions and the experimental dataof Ref. [27], as well as the requirements for translocation, thereby providing insights to the translocation mechanism.

Firstly, and most importantly, both results for the translocation speeds in Eqs. 32 and 35 have the commonfactor −hRflωfl(1 − ρfib) sinα cosα which is multiplying the positive dimensionless expressions Glong and Gshortrespectively. The factor −hωfl gives a directionality for U in agreement with the qualitative prediction of Ref. [27]that CCW rotation will only pull the phage toward the cell body if the phage slides along a right-handed groove.Indeed, our model captures this feature: for right-handed helical wrapping (h = +1) and CCW rotation of the flagellarfilament when viewing the flagellar filament towards the cell body (ωfl < 0), the phage moves towards the cell body(i.e. U > 0).

Secondly, the factor (1− ρfib) reveals that translocation requires anisotropy in the friction between the fibres andthe surface of the flagellar filament (i.e. ρfib 6= 1). We interpret the requirement for anisotropy as an indication ofthe important role of the grooves in guiding the motion of the fibres. The assumption of a helical wrapping of thefibres coupled with this anisotropy simulates the guiding effect of the grooves in this first model by resisting motionperpendicular to the local tangent of the grooves and promoting motion parallel to it.

Thirdly, the presence of the factor sinα cosα shows the requirement of a proper helix i.e. there is no translocationin the limiting cases of a straight (α = 0) or circular wrapping (α = π/2).

Fourthly, translocation requires a non-vanishing value of Lfib in the numerator of both Eqs. 32 and 35. This isbecause the fibres are providing the ‘grip’ by wrapping around the flagellar filament.

Fifthly, the terms involving the tail and head appear in both the numerator and denominator of Eq. 32. Similarly,terms involving the head appear in both the numerator and denominator of Eq. 35. These show that the parts of thephage that are sticking out in the bulk are contributing to both the torque actuating the motion of the phage relativeto the flagellar filament and the drag. In the case of short phages, only the head is providing the torque, hence theterms in the numerator of Eq. 35.

Finally, focusing on the χ-phage, the lengths of the tail and the fibres are similar and the logarithmic dependenceof the resistance coefficients allow us to estimate the fraction in Eq. 32 to be of O(1). The grooves have a pitch ofapproximately 50 nm [27] and the radius of the flagellar filament is approximately 10 nm, giving rise to helix angleα ≈ 51. With ωfl ≈ 100 Hz, we have that U = O(µms−1). Importantly, this means that it is possible for χ-phage totranslocate along a flagellar filament of a few µm long within the timescale of a second, in agreement with the CCWtime interval for bacteria with alternating CCW and CW rotation, thereby enabling the phage to reach the cell bodyand infect the bacterium.

II.5. Dependence of translocation speed on geometrical parameters

We now illustrate the dependence of the translocation speed on the geometrical parameters of the phage, namelythe lengths Lt and Lfib. The asymptotic formulae we obtained above and discussed in §II.4 will help verify theasymptotic behaviour of U for large values of Lt and for vanishing tail length, as well as explain the trends for thetranslocation speed with increasing Lt and Lfib.

To fix ideas, we consider the specific case of the χ phage and hence the following set of parameter values (as in§II.4.1 and Ref. [22]): ah ≈ 30 nm, Lt ≈ 220 nm, rtail = 7 nm, Lfib ≈ 200 nm, rfib ≈ 1 nm, Rfl ≈ 10 nm and

µ = 10−3 Pa.s. For the helix angle we take α = 51 (see §II.4.3). We also take (tx, ty, tz) = (1, 1, 1)/√

3. Notethat for simplicity we do not include the slow variation of the resistive coefficients with Lt, but instead keep their

constant non dimensional values, ζ⊥,t = 4π/ ln(220/7), and ζ⊥,fib = 4π/ ln(4) based on ζ⊥,t ≈ 4πµ/ ln (Lt/rtail) withLt/rtail = 220/7 and ζ⊥,fib ≈ 4πµ/ ln (2d/rfib) with d = 2rfib. We then use Eq. 31 to plot U versus Lt and Lfib

in Fig. 6. For simplicity we have non-dimensionalised lengths by Rfl, time by ω−1fl and viscosity by µ, and denotedimensionless quantities using a hat.

In Fig. 6A we observe that the phage translocation speed is a decreasing function of Lt. The value of U for vanishingtail length is well captured by the theoretical approximation for short-tailed phages of Eq. 36 (red star). For largevalues of Lt, it approaches the theoretical approximation for long-tailed phages (black dash-dotted line, inset). Forthe long-tailed approximation we used the expressions in Eq. A1-A2 in the Supplementary Material (see [35]) keepingterms up to and including L3, for L either Lt or Lfib. We note that the smaller the phage head size, ah, the betterthe convergence between Eq. 31 and the long-tailed approximation (which assumes ah, Rfl Lt, Lfib). In Fig. 6Awe used the dimensions for the χ phage, that correspond to ah/Lfib = 3/22, and the long-tailed approximation also

requires ah Lt. The long-tailed approximation from Eq. 34 captures the decreasing behaviour of U with Lt and

11

FIG. 6. Smooth flagellum model: Dependence of the translocation speed, U , on the geometrical parameters ofthe phage. Left: U vs. the length of the phage tail, Lt, based on Eq. 31 (blue solid line). The speed is a decreasing functionof Lt. The value of U for vanishing tail length is well captured by the theoretical approximation for short-tailed phages (redstar). For large values of Lt it approaches the theoretical approximation for long-tailed phages (black dash-dotted line, inset).Right: U vs. the length of the fibres, Lfib, following Eq. 31. The speed is an increasing function of Lfib. The long-tailedapproximation of Eq. 34 also captures this increasing behaviour.

can be used to explain this result. Specifically, the term involving Lt in the denominator of Eq. 34 shows that thisdecay arises from the resistive part of the hydrodynamic force on the tail (the first term in Eq. 18) which increases asLt increases. In other words, when Lt increases the drag increases and therefore the speed decreases.

In Fig. 6B we next show the speed U as a function of the length of the fibres, Lfib. Clearly the speed is an increasingfunction of Lfib and the long-tailed approximation of Eq. 34 captures this increasing behaviour. We observe thatterms involving Lfib appear in both the numerator and denominator of Eq. 34. Dividing top and bottom by Lfib,we see that as Lfib increases, U increases (at some point U will asymptote to a constant value, but the length atwhich this happens appears to be too large to be relevant biologically). The increase of U with Lfib stems from thepropulsive forces per unit length on the fibres that integrate to a larger propulsive torque as Lfib increases. Of coursethere is also the resistive drag on the fibres but that starts to become more important only at larger values of Lfib.

III. GUIDED TRANSLOCATION ALONG GROOVED FLAGELLAR FILAMENTS

III.1. Geometry

As a more refined physical model, we now include in this section the mechanics arising from the microscopic detailsof the grooved surface of the flagellar filament due to the packing of the flagellin molecules and modify the previouscalculation in order to account for the motion of the phage fibres sliding along the helical grooves.

If the phage slides with speed V along the grooves of helix angle α in the frame of the straight flagellar filament, asillustrated in Fig. 7, then the translocation velocity and rotation rate measured in the laboratory frame, U and ωp,become

U = V cosα, (37)

ωp =hV sinα

Rfl+ ωfl. (38)

12

FIG. 7. Guided translocation of phage along a grooved flagellar filament. The phage, shown in green, has a capsidhead of size 2ah, a tail of length Lt and fibres of cross-sectional radius rfib. The flagellar filament (light blue) has helicalgrooves of helix angle α and is rotating at a rate ωfl. The phage slides along the grooves with speed V in the frame of theflagellar filament. As shown in the inset, the force acting on the fibre sliding along the grooves consists of two parts: (i) a dragresisting the sliding motion of magnitude µV in the −tfib direction and (ii) a restoring force acting to keep the fibre in thecentre of the groove of magnitude kδ in the hbfib direction, where δ is the local offset of the centre of the fibre cross-sectionfrom the centre of the groove and bfib is the local binormal to the fibre centreline that lies in the local tangent plane of thesurface of the flagellar filament and is perpendicular to the tangent of the fibre centreline.

With this substitution we obtain the forces and torques acting on the tail and head as

ez · Ftail = −ζ⊥,tLt

U[1− (1− ρt)t2z

]− ωpRfl(1− ρt)tytz

, (39)

ez ·Mtail = −ζ⊥,tωp

[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)

]− (1− ρt)ty(Utz + ωpRflty)RflLt

, (40)

ez · Fhead = −6πµahU, (41)

ez ·Mhead = −6πµahωp

[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]− 8πµa3hωp. (42)

III.2. Forces and moments

The details of the interactions between the phage fibres and the grooves are expected to be complicated as theydepend on the parts of the flagellin molecules that make up the groove surface and interact with the proteins that thefibres consist of. These interactions could originate from a number of short range intermolecular forces, for exampleelectrostatic repulsion or Van der Waals forces. We model here the resultant of the interaction forces acting on thefibre sliding along the grooves as consisting of two parts, a drag and a restoring force, as shown in the inset of Fig. 7.

Firstly, the fibre is subject to a viscous drag, −µV tfib per unit length where µ is a hydrodynamic resistancecoefficient against the sliding motion (with dimensions of a viscosity). A simple approximation for that coefficientis to assume that there is a fully-developed shear flow resisting the sliding between the fibres and the surface of thegrooves. Assuming the cross-section of the latter to be a circular arc, so that a fraction fcov of the circumference ofa cross-section of the fibres lies inside the groove, we obtain approximately

µ ≈ 2πrfibfcovµ

hgap, (43)

where hgap is the size of the gap between the grooves and the fibres and rfib is the radius of the fibres.Secondly, there should be a restoring force acting to keep the fibre in the centre of the groove arising from the

physical interactions between the fibre and the groove. A simple modelling approach consists of viewing each side

13

of the groove as repelling the fibre, with the resultant of these forces providing a restoring force hkδbfib(s) per unitlength, arising from a potential well 1

2kδ2 where δ is the distance from the centre of the well, and bfib is the local

binormal vector to the fibre centreline,

bfib =

[h cosα sin

(s

Rfl/ sinα

),− cosα cos

(s

Rfl/ sinα

), h sinα

], (44)

that lies in the local tangent plane of the surface of the flagellar filament and is perpendicular to the tangent vectortfib of the fibre centreline. Assuming δ to be uniform along the length of the fibres, the expressions for the force andtorque on the fibres given as the integrals,

Ffib =

∫ L(R)fib

−L(L)fib

−µV tfib(s)− hkδbfib(s) ds, (45)

Mfib =

∫ L(R)fib

−L(L)fib

rfib(s) ∧ [−µV tfib(s)− hkδbfib(s)] ds, (46)

when projected along the z direction become

ez · Ffib = [−µV cosα− kδ sinα]Lfib, (47)

ez ·Mfib = − [µV hRfl sinα− hkδRfl cosα]Lfib. (48)

III.3. Phage translocation: General formulation

The total force and torque balance on the phage along the z-axis,

0 = ez · [Ffib + Ftail + Fhead], 0 = ez · [Mfib + Mtail + Mhead], (49)

give the system to be solved in order to find the two unknown quantities, V and kδ, in terms of ωfl,

0 = [−µV cosα− kδ sinα]Lfib

− ζ⊥,tLt

V cosα

[1− (1− ρt)t2z

]−(hV sinα

Rfl+ ωfl

)Rfl(1− ρt)tytz

− 6πµahV cosα, (50)

0 =− [µV hRfl sinα− hkδRfl cosα]Lfib

− ζ⊥,t(

hV sinα

Rfl+ ωfl

)[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)

]− (1− ρt)ty

[V cosαtz +

(hV sinα

Rfl+ ωfl

)Rflty

]RflLt

− 6πµah

(hV sinα

Rfl+ ωfl

)[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]

− 8πµa3h

(hV sinα

Rfl+ ωfl

). (51)

Using matrix notation, these two equations take the form(A BC D

)(Vkδ

)=

(ZH

)ωfl, (52)

14

where

A =

ζ⊥,tLt

(cosα

[1− (1− ρt)t2z

]− h sinα(1− ρt)tytz

)+ µLfib cosα+ 6πµah cosα

, (53)

B = Lfib sinα, (54)

C =

hµLfibRfl sinα+ ζ⊥,t

h sinα

Rfl

[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)

]− (1− ρt)tyζ⊥,t [cosαtz + h sinαty]RflLt

+ 6πµahh sinα

Rfl

[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]

+ 8πµa3hh sinα

Rfl

, (55)

D = − hRflLfib cosα, (56)

Z = ζ⊥,tLtRfl(1− ρt)tytz, (57)

H = −ζ⊥,t

[LtR

2fl + L2

tRfltx +1

3L3t (t2x + t2y)− (1− ρt)t2yR2

flLt

](58)

+ 6πµah[R2

fl + 2Rfl(Lt + ah)tx + (Lt + ah)2(t2x + t2y)]

+ 8πµa3h

. (59)

The details of the calculation are given in the Supplementary Material (see [35]).Inverting Eq. 52, V and kδ are obtained as

V =DZ −BHAD −BC

ωfl, (60)

kδ =−CZ +AH

AD −BCωfl, (61)

where the full expressions for (DZ − BH),(AD − BC) and (−CZ + AH) for a general phage geometry are given inthe Supplementary Material (see [35]). Finally, from Eq. 60, the translocation velocity along the z-axis is calculatedas U = V cosα.

III.4. Two limits: long vs short-tailed phages

We now proceed by considering the two limiting geometries of long-tailed and short-tailed phages similarly to §II.4.

III.4.1. Long-tailed phages

Under the approximations relevant for long-tailed phages such as χ-phage described in §II.4, i.e. Rfl, ah Lt, Lfib,the translocation velocity along the z-axis gets simplified to

Ulong = −hRflωfl sinα cosαGlong, (62)

Glong =L2t

[13ζ⊥,tLt + 6πµah

](1− t2z)

L2t

[13ζ⊥,tLt + 6πµah

](1− t2z) sin2 α+ µR2

flLfib

· (63)

with the details of the approximation given in the Supplementary Material (see [35]).

III.4.2. Short-tail phages

In the case of short-tail phages, we assume that the tail is negligible and that the fibres are wrapping around theflagellar filament. The translocation velocity simplifies then to

Ushort = −hRflωfl sinα cosαGshort, (64)

Gshort =6πµah

[R2

fl + 2Rflahtx + a2h( 73 − t

2z)]

µLfibR2fl + 6πµah

[R2

fl + 2Rflahtx sin2 α+ a2h sin2 α(73 − t2z

)] · (65)

with all calculation details in the Supplementary Material (see [35]).

15

FIG. 8. Grooved flagellum model: Dependence of the translocation speed on the geometrical parameters of thephage. Left: U as a function of the length of the phage tail, Lt, based on Eq. 60 (blue solid line). The speed is an increasingfunction of Lt. The value of U at vanishing tail length is well captured by the theoretical approximation for short-tailed phages(red star). For large values of Lt, it approaches the theoretical approximation for long-tailed phages (black dash-dotted line).Right: U vs. the length of the fibres, Lfib, based on Eq. 60. The long-tailed approximation of Eq. 63 captures the decreasingbehaviour of U with Lfib.

III.4.3. Interpretation and discussion of the results

Similarly to §II.4.3, we interpret and compare the results in Eqs. 62 and 64. Here again, the crucial factor−hRflωfl sinα cosα appears in both equations multiplying a positive, non-dimensional expression, and we obtainthe correct directionality and speed of translocation in agreement with Ref. [27]. The prefactor −hωfl gives the cor-rect directionality for U , i.e. for right-handed helical wrapping (h = +1) and CCW rotation of the flagellar filament(ωfl < 0), the phage moves towards the cell body (U > 0), in agreement with Ref. [27]. Again, the translocationspeed of O(µm/s) allows translocation during the CCW time interval for bacteria that alternate between CCW andCW sense, in agreement with Ref. [27]. The factor sinα cosα shows that a proper helix is needed for translocation.The presence of the term µR2

flLfib in the denominator implies that the sliding drag from the fibre decreases thetranslocation speed, and longer fibres give a decreased speed. Further, and similarly to §II.4.3, the terms inside thesquare brackets in both the numerator and denominator of Eqs. 63 and 65 show that the parts of the phage that aresticking out in the bulk (for the long phages these are the tail and the head, for the short phages it is only the head),are contributing to both the torque that is actuating the motion of the phage relative to the flagellar filament and tothe drag.

III.5. Dependence of translocation speed on geometrical parameters

We now illustrate the dependence of the translocation speed on the geometrical parameters of the phage, namelyLt and Lfib, according to our model of translocation along grooved flagellar filaments. We use the same approach,parameter values and non-dimensionalisation as in §II.5 and as there we denote dimensionless quantities using a hat.For ˆµ ≡ µ/µ ≈ 2πrfibfcov/hgap, we take rfib/hgap = 2, i.e. assume the gap between the grooves and the fibres ishalf the fibre cross-sectional radius, and fcov = 1/2, so that the cross-section of the grooves is a semi-circle. These

parameter values give ˆµ = 2π. We then use Eq. 60 to plot U (calculated as U = V cosα) as a function of both Lt

16

FIG. 9. Grooved flagellum model: Dependence of the phage translocation speed, U , on the ‘effective’ viscosity,ˆµ (non-dimensionalised by the fluid viscosity µ). As µ increases, the resistive part of the force from the motion of thefibres in the grooves increases, which slows down the phase and leads to a decrease of U .

and Lfib in Fig. 8. In Fig. 8A we observe that the translocation speed is an increasing function of Lt. The value of Uat vanishing tail length is well captured by the theoretical approximation for short-tailed phages of Eq. 65 (red starin figure). For large values of Lt, the result approaches the theoretical approximation for long-tailed phages (blackdash-dotted line).

The long-tailed approximation from Eq. 63, through the terms with Lt in both numerator and denominator, is ableto capture the increasing behaviour of U with Lt. Physically, this trend is caused by the propulsive terms in ez ·Mtail

in Eq. 40 (proportional to L3t ) that increase as Lt increases. Next in Fig. 8B we show the speed U as a function of the

length of the fibres, Lfib, as predicted by Eq. 60, and observe a decreasing trend. The long-tailed approximation ofEq. 63 is able to capture this behaviour. The presence of the term involving Lfib in the denominator of Eq. 63 leadsto a decrease of U with Lfib, and is physically due to an increase of the viscous drag on the fibres as Lfib increases.

Finally, as shown in Fig. 9, we obtain that the translocation speed is a decreasing function of the ‘effective’ viscosityin the grooves, ˆµ, due to the resistive part of the force from the motion of the fibres in the grooves, as expected.

IV. CONCLUSION

In this work, we carried out a first-principle theoretical study of the nut-and-bolt mechanism of phage translocationalong the straight flagellar filaments of bacteria. The main theoretical predictions from our two models, Eqs. 32, 35,62 and 64, give the phage translocation speed, U , in terms of the phage and groove geometries and the rotation rate ofthe flagellar filament, in the two relevant limits of long- and short-tailed phages. These mathematical results capturethe basic qualitative experimental observations and predictions of Refs. [23, 27] for the speed and directionality oftranslocation which are both crucial for successful infection.

The common prefactor in the formulae for the translocation speed along the filament, U ∼ −hωflRfl sinα cosα,appears in the expressions from both models. This provides the expected directionality in agreement with Refs. [23, 27]:sliding of the fibres along flagellar filaments with right-handed helical grooves (h = +1), combined with CCW rotationof flagellar filament (ωfl < 0), will give rise to phage translocation towards the cell body (U > 0) for infection tofollow, whereas CW rotation (ωfl > 0) would translocate the phage away from the cell body (U < 0), towards thefree end of the flagellar filament, and thus away from the cell body.

Quantitatively, the speeds predicted by our model are estimated to a few micrometres per second, U = O(µms−1).This is important for phages infecting bacteria which alternate between CCW and CW senses of rotation. The phageneeds translocation speeds of this magnitude in order to move along a flagellar filament of a few µm long within atimescale of 1 s, which is approximately the CCW time interval [27].

Furthermore, the two limits of long- and short-tailed phages clarify that the physical requirements for translocationalong the flagellar filament are the grip from the part of the phage that is wrapped around the flagellar filament,in a helical shape (indeed as dictated by the shape of the grooves), combined with torque provided by the partsof the phage sticking out in the bulk, away from the flagellar filament. The plots for the translocation speed as afunction of the phage tail length approach the asymptotic approximations for long- and short- tailed phages at largeand vanishing tail lengths respectively in both models.

17

The important point where the two models deviate from each other is their opposite predictions for the translocationspeed as a function of the phage tail length and the phage fibre length. We conjecture that the second model withits explicit inclusion of the grooves should be closer to the real-life situation. According to it, U increases when Lt

increases because the propulsive terms in the axial torque on the tail increase with Lt. In contrast, the decreasingbehaviour of U with Lfib is caused by the resistive part of the force exerted by the motion of the fibres in the groovesthat increases with Lfib, and thus slows down the motion.

Having modelled phage translocation along straight flagellar filaments of mutant bacteria, the next natural stepwill be to model phage translocation along the naturally helical flagellar filaments of wild type bacteria. In this case,the geometry is more complicated as it involves motion along helical grooves on top of a flagellar filament whosecentreline is also a helix. This is a more complicated system geometrically, as the helical fibres are sliding along ahelical flagellar filament with a spatially-varying local tangent and therefore we expect that numerical computationswould be required in order to tackle it. Notably, there will be an additional hydrodynamic drag on the phage dueto the rotation and translation of the flagellar filament. The hydrodynamic drag from the translation will oppose orenhance the translocation of the phage towards the cell body depending on the chirality of the flagellar filament. Thisopens up the possibility of a competition between the nut-and-bolt translocation effect and the possibly opposingdrag due to translation, which will vary with the helical angle of the flagellar filament. Different regimes are expectedto arise as the helical angle of the flagellar filament is increased from 0, hinting to a rich, nonlinear behaviour to beinvestigated with the aid of numerical simulations.

In this work, we focus on the translocation of the phage once it has reached a steady, post-wrapping state, and thusassuming that it is moving rigidly. Future studies could address the transient period of wrapping, where the lengthof the fibres wrapped around the filament is increasing and the ‘grip’ is possibly becoming tighter. Some phages (inthe Siphoviridae family) have long, flexible tails, thereby requiring the addition of the elasticity of the tail and fibresinto the model. We hope that the modelling developed in this paper will motivate not only further theoretical studiesalong those lines but also more experimental work clarifying the processes involved in the wrapping and motion ofthe fibre in the grooves.

ACKNOWLEDGEMENTS

This work was funded by the EPSRC (PK) and by the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreement 682754 to EL).

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